About 60 world leaders are taking part in the one-day meeting organised by UN Secretary General António Guterres.
He earlier said countries could only speak at the summit if they came with action plans to cut carbon emissions.
Brazil and Saudi Arabia are among the countries staying away.
The summit comes days after several million people took part in a global climate strike led by youth activists.
Ahead of the meeting, scientists warned that the signs and impacts of global warming were speeding up.
The World Meteorological Organization (WMO) said the amount of carbon dioxide going into the atmosphere between 2015 and 2019 had grown by 20% compared with the previous five years.
“We should listen to the loud cry coming from the schoolchildren,” said Professor Brian Hoskins, chair of the Grantham Institute, Imperial College London, and professor of meteorology at the University of Reading.
“There is an emergency – one for action in both rapidly reducing our greenhouse gas emissions towards zero and adapting to the inevitable changes in climate,” he said.